A320067 Expansion of Product_{k>0} theta_3(q^k), where theta_3() is the Jacobi theta function.

1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 78, 106, 152, 202, 258, 370, 478, 602, 828, 1042, 1332, 1758, 2198, 2758, 3572, 4448, 5518, 7012, 8636, 10654, 13350, 16362, 19946, 24722, 30070, 36478, 44776, 54010, 65202, 79234, 95196, 114166, 137686, 164530, 196252, 235308, 279718, 332002

Offset

0,2

Comments

Also the number of integer solutions (a_1, a_2, ..., a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

Expansion of Product_{k>0} eta(q^(2*k))^5 / (eta(q^k)*eta(q^(4*k)))^2.

a(n) ~ log(2)^(3/8) * exp(Pi*sqrt(n*log(2))) / (4 * Pi^(1/4) * n^(7/8)). - Vaclav Kotesovec, Oct 05 2018

Expansion of Product_{k>0} theta_4(q^(2*k))/theta_4(q^(2*k-1)), where theta_4() is the Jacobi theta function. - Seiichi Manyama, Oct 26 2018

Mathematica

nmax = 50; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)

Prog

(PARI) m=50; x='x+O('x^m); Vec(1/(prod(k=1, 2*m, prod(j=1, floor(2*m/k), (1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2 )))) \\ G. C. Greubel, Oct 29 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2: j in [1..Floor(2*m/k)]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018

Crossrefs

Cf. A000122, A029594, A033715, A320068, A320078, A320139, A320968, A320992.

Sequence in context: A320246 A320247 A320248 * A248823 A284616 A136513

Adjacent sequences: A320064 A320065 A320066 * A320068 A320069 A320070

Keyword

nonn,nice

Author

Seiichi Manyama, Oct 05 2018

Status

approved